In a PET (“Positron Emission Tomography”) scan, a radioactive material is introduced into the patient's body. As the radioactive material decays, it emits positrons. When one of these positrons meets an electron, the two particles annihilate, leaving behind a pair of gamma ray photons that travel in opposite directions. The line along which the gamma ray photons travel is referred to as a “line of response.” The location at which the annihilation occurred is referred to as an “annihilation site.” Throughout this disclosure, the term “gamma ray” will refer to gamma ray photons.
A PET scanner includes a large number of scintillation crystals. When a gamma ray strikes one of these scintillation crystals, it causes the generation of photons. These photons are detected by photodetectors associated with each scintillation crystal. The detection of such a photon by a photodetector is referred to as an “event.” When two photodetectors detect an event at approximately the same time, that event is referred to as a “coincidence.”
The scintillation crystals are arranged in crystal columns arranged side-by-side in a ring around the patient. Each crystal column includes a number of crystals arranged in a line that extends in a direction parallel to an axis of the patient. This direction is referred to as the “axial direction.” Any direction perpendicular to the axial direction is referred to as the “trans-axial” direction. The location of a particular scintillation crystal is thus defined by a trans-axial coordinate, which identifies the particular column that it belongs to, and an axial coordinate, which identifies where in the column the crystal is actually located.
To reconstruct an image, one divides the volume of interest into voxels. For each such voxel, and for each pair of crystals, there exists a probability that an annihilation at that voxel will result in a pair of gamma rays that proceed along a line of response that ultimately intersects that pair of scintillation crystals. The set of all these probabilities for all voxels and all crystal pairs defines a “geometric system response matrix” with a number of rows corresponding to the number of voxels and a number of columns corresponding to the number of crystal pairs (or vice versa).
The distribution of events in the crystals can be represented by a “crystal vector” having as many elements as there are crystal pairs. The value of each element in the crystal vector provides information on an event detected at the crystal pair corresponding to that element. In the case of the geometric system response matrix, each element provides information on the intensity with which annihilation gamma-ray pairs strike the crystal pair corresponding to that element. In the case of the full system response matrix, each element provides information on the intensity with which annihilation gamma-pairs are detected by the crystal pair corresponding to that element.
The volume of interest, or the “emission source volume”, can be represented as a “voxel vector” having as many elements as there are voxels. The value of each element in the voxel vector indicates the occurrence of an annihilation at that voxel. These values are, of course, unknown. It is these values that an image-reconstruction method seeks to estimate.
In a geometric forward projection, the crystal vector is obtained by multiplying the voxel vector by the geometric system response matrix. The full system response matrix can be factorized into the product of a geometric system response matrix with three matrices, each of which maps from the crystal vector back into the same crystal vector space. The first two such matrices are diagonal matrices that contain the probability of gamma-ray pair attenuation within the emission source volume, and the probability of detecting a gamma-ray pair striking a particular detector pair, respectively. The third such matrix is a non-diagonal blurring matrix whose elements give the probability that a gamma-ray pair striking a particular crystal pair will be detected by that crystal pair or by nearby crystal pairs. The geometric system response matrix is by far the largest of these matrices. Multiplication of the source vector by this matrix is thus the most computationally intensive part of a forward projection. To simplify the subsequent discussion, the full system response matrix will be treated as being equivalent to geometric system response matrix below.
In principle, therefore, the values in the voxel vector, and therefore the most likely spatial distribution of annihilation sites that led to an observed crystal vector, can be determined by simply multiplying the crystal vector by the inverse of the system response matrix. The crystal vector is known, because it is this vector that is measured. The geometric system response matrix depends only on geometry, and therefore its inverse could in principle be pre-calculated and stored. In practice, however, this direct inversion procedure is made difficult by intrinsic fluctuations due to limited event statistics. As an alternative, iterative image reconstruction algorithms have been developed that make explicit use of the system response matrix in each iteration.
For a practical PET scanner, the system response matrix is far too large for either direct inversion or storage. For example, a typical PET scanner can have on the order of 200 million pairs of crystals. The voxels in a typical region of interest can be 2 mm cubes arranged in a three-dimensional array having a 256×256 voxel face and extending 172 voxels deep. This results in approximately 11 million voxels. The resulting System Response Matrix would thus have approximately 2.44 quintillion elements (2.47×1015). Such a large matrix is presently impractical to store.